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Factoring quadratics is a crucial step in solving quadratic equations without resorting to the quadratic formula. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The goal of factoring is to break down the equation into simpler terms that can be set equal to zero and solved for the variable.

To factor a quadratic equation, one generally looks for two numbers that multiply to give the product of the coefficient of \( x^2 \) (a) and the constant term (c), and at the same time, add up to the coefficient of \( x \) (b). In the example \((x-2)(x+3)=0\), the equation is already factored for us. The two binomials, \((x-2)\) and \((x+3)\) represent the factors. In this case, the numbers -2 and 3 multiply to give -6 (the product of the coefficient of \(x^2\) and the constant term when the equation is expanded) and add up to 1, which is the coefficient of \(x\) in the expanded form \(x^2 + x - 6 = 0\).

The zero product property is a key rule in algebra stating that if the product of two factors is zero, then at least one of the factors must be zero. This rule allows us to solve quadratic equations once we have factored them. After factoring, each factor is set equal to zero independently, as any one of them being zero would satisfy the equation.

Applying this property to the problem \((x-2)(x+3)=0\), we get two separate equations, \((x-2) = 0\) and \((x+3) = 0\). These equations can be solved individually to find the values of \(x\) that make the original equation true. The zero product property simplifies the process of finding solutions to quadratics, making it a powerful tool when paired with factoring.

Quadratic solutions are the values of the variable that satisfy the quadratic equation. For the given equation \((x-2)(x+3)=0\), we found two solutions using the factoring and zero product property methods: \(x = 2\) and \(x = -3\). Each of these solutions is the result of setting the factored forms equal to zero and solving for \(x\).

Every quadratic equation has two solutions, which can be real or complex numbers. The solutions found through factoring can quickly provide the real number solutions. However, when a quadratic cannot be factored easily, other methods like completing the square or using the quadratic formula are required to find the solutions. These two solutions are crucial not just for the equation itself, but they also represent the x-intercepts or the points where the graph of the quadratic equation crosses the x-axis.